In order to interpret why we have arrived at this end result, we find it useful to engage in a high level discussion of the generic closure problem, motivated by, for example, [1]} and [2]}, and in text books, e.g. [3]}. In the context of Hamilton's equation for the time evolution of \(a_{{\bf p}}\) , one multiplies by \(a_{{\bf p}}^{\ast }\) , multiplies the complex conjugate of Hamilton's equation by \(a_{{\bf p}}\) , subtracts the two equations and then averages to obtain an evolution equation for the second order wave action \(n_{{\bf p}} \)  in terms of the third order correlation function. The process continues iteratively, deriving an equation for the third order correlation function that involves fourth order correlations, building up a heirarchy of unclosed equations. If we eliminate all subscripts, coefficients and summations, the structure can be schematically represented as
